Lozenge Land

Read a flat tiling as deep stone, and cut the cheapest stairs that let the water climb it.

The Lozenge Land board

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Grades
3–6
Subjects
Shapes & Space, Logic
Kind
Puzzle
Cost
Free, no account, no ads

What it is

The board is a tiling of three kinds of rhombus, and it is also a heap of stone blocks seen from one corner: pale floors, a sunlit wall and a shaded one. Tap a corner where all three kinds meet and a block pops out, or back in. Water starts on the lowest floor and climbs any step of one block, but a drop of two is a cliff it cannot get up. Every floor it reaches turns to grass. Cut stairs, or shave stone away, until the whole land is green, changing no more blocks than the aim allows.

The rules

Cut stairs until the water reaches every floor.

A small yard of stone with a tower of two blocks in the corner. Dots mark the corners where all three kinds of face meet. A finger taps the one on top of the tower and its top block lifts away; it taps another and a block sets itself down on the floor.
Tap a corner where all three kinds of face meet: a block pops out of the stone, or back in.
A stone cut in three steps. Water rises from the lowest floor and spreads, turning that floor and the terrace one block above it to grass. The top terrace, two blocks up, stays dry stone.
Water starts on the floor nearest you and climbs a step of one block. Two blocks is a cliff, and it stops.
A terrace of stone standing two blocks over a green floor, too high for the water. One block laid at its foot makes a step, the water climbs it, the whole land turns green and a goat walks up. Then the same land again, watered instead by taking one block off the top of the terrace.
Build a step, or shave one off — in as few blocks as the aim says.

Every aim is the true best: no carving of the stone waters it in fewer.

What it gives you to practise

Seeing depth in a flat drawing, and holding on to it. The same rhombuses read as a heap or as a hollow, and the trees standing on the floors are there to say which way is up. Then planning: a block can only sit where the blocks behind it and beside it hold it up, so the cheapest stair is rarely the first one that comes to mind. On every board the aim is the true minimum, proved by the computer before the stone is shown — the random stones too — so matching it is a real result.

In a classroom

The board prints as line art with the grass taken off, so finding where the water stops becomes the paper puzzle and a pencil marks the blocks to change. The first stones in Level I need a block or two; a couple of those, before anybody tries a ten-across stone, is enough for a class to agree on which way is up.

Share or embed

The whole puzzle at one address, with every board in it. Whoever opens it stays inside it, and their progress stays in their own browser.

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